WASHINGTON — A top U.S. counterterrorist official says there are “a number of specific threats” aimed at the Winter Olympics that start this week in Sochi, Russia — with the greatest danger coming from the Caucasus Emirate, which has threatened to attack the games.

Matthew Olsen, director of the National Counterterrorism Center, told Congress on Tuesday that the U.S. and Russia are tracking threats of “varying degrees” of credibility. He said potential attacks seem more likely outside the venues for the games, and instead in the area or region around Sochi.

He described intelligence sharing with Russia as “good.”

FBI Director James Comey said U.S. cooperation with the Russian Security Service has improved.

Russia has mounted a massive security operation for the Olympics, deploying more than 50,000 police and soldiers amid threats from Muslim insurgents.

Top U.S. security officials told President Barack Obama that all appropriate steps are being taken to keep Americans safe.

The White House said Obama received an update on security for the Olympics on Tuesday. Vice President Joe Biden, Secretary of State John Kerry and CIA Director John Brennan were among the officials taking part. Leaders from the FBI and the Pentagon also joined.

The White House said Obama directed his team to work closely with Russia and others to make the Sochi Games safe.